A new over-penalized weak Galerkin finite element method. II: Elliptic interface problems
type: Item
instance of: scholarly article
title: A new over-penalized weak Galerkin finite element method. II: Elliptic interface problems
zbMATH Open document ID: 1482.65215
author: Wenya Qi
author: Kaifang Liu
author: Qingxian Gu
author: Lunji Song
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2021-06-17T00:00:00Z
review text: The goal of this article is to introduce and analyse a mesh-based numerical method for computing the solution to discontinuous elliptic interface problems. Interface problems involving non-homogenous boundary conditions and jumping coefficients such as the one considered here can be slightly delicate to solve and analyse -- one reason being the lack of global solution regularity -- although there is now a considerable literature on this topic (see, e.g., [\textit{J. Guzmán} et al., Math. Comput. 85, No. 301, 2071--2098 (2016; Zbl 1342.65218)] or [\textit{X. Claeys} et al., Adv. Comput. Math. 43, No. 5, 1075--1101 (2017; Zbl 1381.65094)] for an approach based on boundary integral equations and the references therein). The specific problem considered by the authors in this paper is the following: Given a polygonal domain \(\Omega \subset \mathbb{R}^2\) that can be partitioned as \(\Omega= \Omega_1 \cup \Omega_2\) with \(\Omega_1, \Omega_2 \subset \mathbb{R}^2\) being Lipschitz and having common interface \(\Gamma:= \overline{\Omega_1} \cap \overline{\Omega_2}\), we seek to the solution to the interface problem: \[ \begin{split} -\nabla \cdot (A \nabla u)&=f \qquad \text{in } \Omega,\\ u&=g \qquad \text{on } \partial \Omega \setminus \Gamma,\\ \llbracket u \rrbracket_{\Gamma} &=\psi \qquad \text{on } \Gamma,\\ \llbracket A \nabla u \cdot \boldsymbol{\eta}\rrbracket_{\Gamma}&=\phi \qquad \text{on } \Gamma. \end{split}\tag{1} \] Here, \(A=A(x) \colon \Omega \rightarrow \mathbb{R}^{2 \times 2}\) is a symmetric matrix-valued function that is assumed positive definite and piecewise constant (w.r.t. \(\Omega_1\) and \(\Omega_2\)). Moreover \(\llbracket u \rrbracket_{\Gamma}\) and \(\llbracket A\nabla u \cdot \eta\rrbracket_{\Gamma}\) denote the usual Dirichlet jump and co-normal derivative jumps of \(u \in H^1(\Omega)\) respectively, i.e., \begin{align*} \llbracket u\rrbracket_{\Gamma}:=& u\vert_{\partial \Omega_1} + u \vert_{\partial \Omega_2},\\ \llbracket A\nabla u \rrbra…
reviewed by: Muhammad Hassan
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 65N15
Mathematics Subject Classification ID: 35J50
zbMATH DE Number: 7360477
zbMATH Keywords: elliptic interface problems
zbMATH Keywords: over-penalized weak Galerkin
zbMATH Keywords: double-valued function
zbMATH Keywords: error estimates
MaRDI profile type: Publication
full work available at URL: dcdsb.2020196
OpenAlex ID: W3034774960
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DOI: 10.3934/DCDSB.2020196
description: scientific article; zbMATH DE number 7360477